Question

Find the value is of a and b, so that (x+1) and (x-1) are factors of x⁴+ax³-3x²+2x+b.

Answer 1

$\textbf{Concept used:}$

$\textbf{Factor theorem:}$

$\text{(x-a) is a factor of f(x) iff f(a) =0}$

$\text{Let f(x)=}x^4+ax^3-3x^2+2x+b$

$\text{Since (x-1) is a factor of f(x), f(1)=0}$

$\implies\;1^4+a(1)^3-3(1)^2+2(1)+b=0$

$\implies\;1+a-3+2+b=0$

$\implies\;a+b=0$.......(1)

$\text{Since (x+1) is a factor of f(x), f(-1)=0}$

$\implies\;(-1)^4+a(-1)^3-3(-1)^2+2(-1)+b=0$

$\implies\;1-a-3-2+b=0$

$\implies\;-a+b=4$.......(2)

$\text{Adding (1) and (2), we get}$

$2b=4$

$\implies\boxed{\bf\;b=2}$

$\text{Put b=2 in (1), we get}$

$a+2=0$

$\boxed{\bf\;a=-2}$

Find more:

Find the value is of a and b, if x²-4 is a factor of ax⁴+2x³ -3x²+bx-4.

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